3.2834 \(\int \frac {1}{\sqrt {\frac {c}{(a+b x)^3}}} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]

[Out]

2/5*(b*x+a)/b/(c/(b*x+a)^3)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c/(a + b*x)^3],x]

[Out]

(2*(a + b*x))/(5*b*Sqrt[c/(a + b*x)^3])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\frac {c}{(a+b x)^3}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {c}{x^3}}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c/(a + b*x)^3],x]

[Out]

(2*(a + b*x))/(5*b*Sqrt[c/(a + b*x)^3])

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fricas [B]  time = 0.82, size = 79, normalized size = 3.16 \[ \frac {2 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{5 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c/(b*x+a)^3)^(1/2),x, algorithm="fricas")

[Out]

2/5*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)
)/(b*c)

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giac [B]  time = 0.36, size = 144, normalized size = 5.76 \[ \frac {2 \, {\left (15 \, \sqrt {b c x + a c} a^{2} - \frac {10 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a}{c} + \frac {15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}}{c^{2}}\right )}}{15 \, b c \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c/(b*x+a)^3)^(1/2),x, algorithm="giac")

[Out]

2/15*(15*sqrt(b*c*x + a*c)*a^2 - 10*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a/c + (15*sqrt(b*c*x + a*c
)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c + 3*(b*c*x + a*c)^(5/2))/c^2)/(b*c*sgn(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*
x + a^3)*sgn(b*x + a))

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maple [A]  time = 0.00, size = 22, normalized size = 0.88 \[ \frac {\frac {2 b x}{5}+\frac {2 a}{5}}{\sqrt {\frac {c}{\left (b x +a \right )^{3}}}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/(b*x+a)^3*c)^(1/2),x)

[Out]

2/5*(b*x+a)/b/(1/(b*x+a)^3*c)^(1/2)

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maxima [A]  time = 0.56, size = 27, normalized size = 1.08 \[ \frac {2 \, {\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{5 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c/(b*x+a)^3)^(1/2),x, algorithm="maxima")

[Out]

2/5*(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^(3/2)/(b*c)

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mupad [B]  time = 1.25, size = 26, normalized size = 1.04 \[ \frac {2\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,{\left (a+b\,x\right )}^4}{5\,b\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c/(a + b*x)^3)^(1/2),x)

[Out]

(2*(c/(a + b*x)^3)^(1/2)*(a + b*x)^4)/(5*b*c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {c}{\left (a + b x\right )^{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c/(b*x+a)**3)**(1/2),x)

[Out]

Integral(1/sqrt(c/(a + b*x)**3), x)

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